3.842 \(\int \frac{1}{x^4 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=79 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{3 a x^3} \]

[Out]

-Sqrt[a - b*x^4]/(3*a*x^3) + (b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1
/4)*x)/a^(1/4)], -1])/(3*a^(3/4)*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.0627656, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(3*a*x^3) + (b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1
/4)*x)/a^(1/4)], -1])/(3*a^(3/4)*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 9.0855, size = 66, normalized size = 0.84 \[ - \frac{\sqrt{a - b x^{4}}}{3 a x^{3}} + \frac{b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{3 a^{\frac{3}{4}} \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-b*x**4+a)**(1/2),x)

[Out]

-sqrt(a - b*x**4)/(3*a*x**3) + b**(3/4)*sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1
/4)*x/a**(1/4)), -1)/(3*a**(3/4)*sqrt(a - b*x**4))

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Mathematica [C]  time = 0.256946, size = 90, normalized size = 1.14 \[ \frac{-\frac{i b \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}}-\frac{a}{x^3}+b x}{3 a \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a - b*x^4]),x]

[Out]

(-(a/x^3) + b*x - (I*b*Sqrt[1 - (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sq
rt[a])]*x], -1])/Sqrt[-(Sqrt[b]/Sqrt[a])])/(3*a*Sqrt[a - b*x^4])

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Maple [A]  time = 0.018, size = 88, normalized size = 1.1 \[ -{\frac{1}{3\,a{x}^{3}}\sqrt{-b{x}^{4}+a}}+{\frac{b}{3\,a}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-b*x^4+a)^(1/2),x)

[Out]

-1/3*(-b*x^4+a)^(1/2)/a/x^3+1/3*b/a/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(
1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2
)*b^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^4),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-b x^{4} + a} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^4),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-b*x^4 + a)*x^4), x)

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Sympy [A]  time = 2.8603, size = 42, normalized size = 0.53 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-b*x**4+a)**(1/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*x*
*3*gamma(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^4), x)